- Inicio
- Vectors
- Product of a real number by a vector
- Ejercicios
Product of a real number by a vector
Given the vectors $\vec{u}=(2,-2)$ and $\vec{v}=(1,3)$, determine:
- $3\vec{u}-2\vec{v}$
- $-\vec{u}-\vec{v}$
- $5\vec{u}+2\vec{v}$
- $\vec{u}+3\vec{v}$
Is there one which is a unit vector?
- $3\vec{u}-2\vec{v}=3(2,-2)-2(1,-3)=(6,-6)+(-2,6)=(4,0)$
- $-\vec{u}-\vec{v}=-(2,-2)-(1,-3)=(-3,5)$
- $5\vec{u}+2\vec{v}=5(2,-2)+2(1,-3)=(10,-10)+(2,-6)=(12,-16)$
- $\vec{u}+3\vec{v}=(2,-2)+3(1,-3)=(5,-11)$
$\begin{array}{l} |(4,0)|=\sqrt{4^2+0^2}=\sqrt{16}=\sqrt{4^2}=4 \ |(-3,5)|=\sqrt{(-3)^2+5^2}=\sqrt{9+25}=\sqrt34 \ |(12,-16)|=\sqrt{12^2+(-16)^2}=\sqrt{144+256}=\sqrt{400}=\sqrt{20^2}=20 \ |(5,-11)|=\sqrt{5^2+(-11)^2}=\sqrt{25+121}=\sqrt{146} \end{array} $
We can see, then, that none of these norms is one. Therefore, none of these vectors are unit vectors.
- $(4,0)$
- $(-3,5)$
- $(12,-16)$
- $(5,-11)$
None of these vectors is a unit vector.